Systems of Linear Equations in Two Variables – Graphing

March 8, 2011 at 2:54 pm Leave a comment

There are 3 methods for solving systems of two linear equations in two variables: by graphing, the substitution method, and the addition (or elimination) method. Without using technology, the least efficient technique is solving by graphing. Even if a student scales their graph perfectly, if the intersection of the two lines does not occur at a point whose coordinates are integers we cannot determine the exact solution. (Never mind if the student’s graphs are not perfect.) So, why even teach students to solve systems by graphing?

I start to teach systems by graphing because it helps students develop a conceptual understanding of what a solution represents. Suppose we had the following system of equations:

3x+2y=12\\y=-\frac {1}{6} x-2

When we graph the line 3x+2y=12, we are displaying the ordered pairs that are solutions to that equation. When we graph the line y=-\frac {1}{6} x-2, we are displaying the ordered pairs that are solutions to that equation. So, what does the point of intersection represent? This is the one ordered pair that is a solution to both equations. This helps students to understand what the solution of a system of equations represents.

Once we understand how to find the solution of an independent system of equations by graphing, I ask my students if all systems of linear equations will always have one solution. Someone will always ask “What happens if the two lines are parallel?” Since distinct parallel lines do not intersect, my students understand that some systems will have no solution. (We call this type of system an inconsistent system.)

I then ask if there are any other options besides one solution and no solution. Can two lines have more than one point in common? We then figure out that this could happen only if the two lines are identical. Now my students understand the type of system we call a dependent system. By the way, since graphing lines is so fresh in their minds, along with the slope-intercept form of a line, my students have an easier time understanding how to present solutions to a dependent system as an ordered pair of the form (x, mx+b).

Once I am sure that students understand the graphical interpretations associated with these types of systems, I then move on to the substitution method and later to the addition method. But this discussion on solving by graphing is invaluable to student understanding.

By the way, I do not ask students to solve systems by graphing on their exam. I do ask questions like the following:

  • Draw two lines that would form an inconsistent system of equations.
  • Draw two lines that would form a system whose only solution is (3,8).
  • Draw two lines that would form a dependent system of equations.
    [Yes, essentially I’m finding out if they can trace a line 🙂 ]

How do you introduce this topic? Do you ignore it? Do you incorporate technology? Please leave a comment, or reach me through the contact page at my web site –


I am a math instructor at College of the Sequoias in Visalia, CA. If there’s a particular topic you’d like me to address, or if you have a question or a comment, please let me know. You can reach me through the contact page on my website –


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